In this case, la\boldsymbol l_{a} is the Lie derivative along the vector fieldaa, and the conditions i[a,b]=[ia,lb]\boldsymbol i_{[a,b]}= [\boldsymbol i_{a}, \boldsymbol l_{b}] and [ia,ib]=0[\boldsymbol i_{a}, \boldsymbol i_{b}]=0, together with the defining equation la=[dΩX*,ia]\boldsymbol l_{a}=[d_{\Omega ^{*}_{X}},\boldsymbol i_{a}] and with the equations l[a,b]=[la,lb]\boldsymbol l_{[a,b]}=[\boldsymbol l_{a},\boldsymbol l_{b}] and [dΩX*,la]=0[d_{\Omega ^{*}_{X}},\boldsymbol l_{a}]=0 expressing the fact that l:𝒯X→ℰnd*(ΩX*)\boldsymbol l\colon \mathcal{T}_{X}\to \mathcal{E}nd^{*}(\Omega ^{*}_{X}) is a dglamorphism, are nothing but the well-known Cartan identities involving contractions and Lie derivatives.